Quesiton number 3 and number 4. For quesiton number 4 extra credit one, make sure you use matlab to do, Include the matlab code and the result in the solution.
MATH-4500 Spring 2023 PDEs of Math/Phys Problem Set 5 D.W. Schwendeman Notes 1. DUE DATE: Monday, March 20, by 10am (start of class). 2. Please show all work for the problems. Illegible or undecipherable solutions will not be graded. 3. Figures, if any, should be displayed clearly and properly labelled. Problems 1. Consider the regular Sturm-Liouville eigenvalue problem( xφ′ )′ + λxφ = 0, 1 < x="">< 2,="" φ(1)="φ(2)" =="" 0="" (a)="" use="" the="" rayleigh="" quotient="" to="" show="" that="" λ=""> 0 . (Hint: first show that λ ≥ 0 , and then consider the case λ = 0 separately.) (b) Use the Rayleigh quotient and a test function of the form u(x) = a+ bx−x2 , with suitably chosen constants a and b , to estimate the value of the smallest eigenvalue, λ1 . (c) Extra credit. Consider a new test function v(x) = (1 + cx)u(x) , where c is a constant and u(x) is the test function in part (b). Note that c = 0 gives the same estimate for λ1 in part (b). However, if c 6= 0 , then v(x) may give an improved estimate for λ1 using the Rayleigh quotient. Find the value for c that gives the best estimate. (Notes: The algebra involved for this problem is messy and I used Maple to help solve it. Also, you could read ahead in section 7.7.7 of the textbook to work out an exact formula to find λ1 so that you can compare your estimates with the exact value.) 2. The vertical displacement of a vibrating rectangular membrane satisfies utt = c 2(uxx + uyy), 0 ≤ x ≤ L, 0 ≤ y ≤ H, t ≥ 0 with boundary conditions u(0, y, t) = u(L, y, t) = uy(x, 0, t) = uy(x,H, t) = 0 and initial conditions u(x, y, 0) = 0, ut(x, y, 0) = g(x, y) Use separation of variables to find the solution u(x, y, t) . Determine the natural frequencies of vibra- tion. (Hint: start with u(x, y, t) = φ(x, y)T (t) .) 3. Consider the heat flow in a three-dimensional region R with smooth boundary S and unit outward normal n . The temperature u(x, t) satisfies ut = k∇2u+Q(x), x ∈ R, t > 0, where k > 0 is a constant thermal diffusivity and Q(x) is a heat source, both assumed to be known. The initial and boundary conditions are u(x, 0) = f(x), x ∈ R n · ∇u = g(x), x ∈ S 1 (a) Define the total thermal energy in the region as E(t) = ∫ R u(x, t) dV . Integrate the PDE, and use the divergence theorem, to determine a formula for ddtE(t) . (b) Determine a condition on Q(x) and g(x) such that a steady state solution exists. (You need not find this solution.) Give a brief physical interpretation of the condition you derive. (c) Assuming the condition in part (b) holds, determine E(t) . 4. Determine expressions for the eigenvalues λ and eigenfunctions φ(r) for the following Sturm-Liouville eigenvalue problems. (a) ( rφ′ )′ + ( λr − 4 r ) φ = 0, 0 < r="">< 3,="" φ(0)="" bounded="" and="" φ′(3)="0" (b)="" (="" rφ′="" )′="" +="" (="" λr="" −="" 1="" r="" )="" φ="0," 1="">< r="">< 2, φ′(1) = 0 and φ(2) = 0 extra credit. use maple’s besselj and bessely functions (or matlab’s besselj and bessely) to de- termine numerical values for the smallest three eigenvalues for each problem, and plot the corresponding three eigenfunctions. 2 0321905652.pdf applied partial differential equations with fourier series and boundary value problems this page intentionally left blank applied partial differential equations with fourier series and boundary value problems fifth edition richard haberman southern methodist university boston columbus indianapolis new york san francisco upper saddle river amsterdam cape town dubai london madrid milan munich paris montreal toronto delhi mexico city sao paulo sydney hong kong seoul singapore taipei tokyo editor in chief: deirdre lynch cover designer: bruce kenselaar senior acquisitions editor: william hoffman manager, cover visual research and associate editor: brandon rawnsley permissions: jayne conte executive marketing manager: jeff weidenaar cover art: artida/fotolia marketing assistant: caitlin crain full-service project management: integra senior production project manager: beth houston composition: integra credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within text. copyright c© 2013, 2004, 1998, 1987, 1983 by pearson education, inc. all rights reserved. manufactured in the united states of america. this publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. to obtain permission(s) to use material from this work, please submit a written request to pearson education, inc., permissions department, one lake street, upper saddle river, new jersey 07458, or you may fax your request to 201-236-3290. many of the designations by manufacturers and sellers to distinguish their products are claimed as trademarks. where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. library of congress cataloging-in-publication data haberman, richard, applied partial differential equations : with fourier series and boundary value problems / richard haberman. – 5th ed. p. cm. isbn-13: 978-0-321-79705-6 (alk. paper) isbn-10: 0-321-79705-1 (alk. paper) 1. differential equations, partial. 2. fourier series. 3. boundary value problems. i. title. qa377.h27 2013 515’.353–dc23 2012013033 10 9 8 7 6 5 4 3 2 1 cw 16 15 14 13 12 isbn-10: 0321797051 isbn-13: 9780321797056 to liz, ken, and vicki this page intentionally left blank contents preface xvii 1 heat equation 1 1.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 derivation of the conduction of heat in a one-dimensional rod . . . . 2 1.3 boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 equilibrium temperature distribution . . . . . . . . . . . . . . . . . . . 14 1.4.1 prescribed temperature . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 insulated boundaries . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 derivation of the heat equation in two or three dimensions . . . . . . 19 appendix to 1.5: review of gradient and a derivation of fourier’s law of heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 method of separation of variables 32 2.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 heat equation with zero temperatures at finite ends . . . . . . . . . 35 2.3.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 separation of variables . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 time-dependent ordinary differential equation . . . . . . . . 37 2.3.4 boundary value (eigenvalue) problem . . . . . . . . . . . . . 38 2.3.5 product solutions and the principle of superposition . . . . . 43 2.3.6 orthogonality of sines . . . . . . . . . . . . . . . . . . . . . . 46 2.3.7 formulation, solution, and interpretation of an example . . . 48 2.3.8 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 appendix to 2.3: orthogonality of functions . . . . . . . . . . . . . . . . . . . 54 2.4 worked examples with the heat equation (other boundary value problems) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4.1 heat conduction in a rod with insulated ends . . . . . . . . 55 2.4.2 heat conduction in a thin insulated circular ring . . . . . . 59 2.4.3 summary of boundary value problems . . . . . . . . . . . . . 64 2.5 laplace’s equation: solutions and qualitative properties . . . . . . . . 67 2.5.1 laplace’s equation inside a rectangle . . . . . . . . . . . . . 67 2.5.2 laplace’s equation inside a circular disk . . . . . . . . . . . 72 2.5.3 fluid flow outside a circular cylinder (lift) . . . . . . . . . 76 2.5.4 qualitative properties of laplace’s equation . . . . . . . . . . 79 vii viii contents 3 fourier series 86 3.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2 statement of convergence theorem . . . . . . . . . . . . . . . . . . 2,="" φ′(1)="0" and="" φ(2)="0" extra="" credit.="" use="" maple’s="" besselj="" and="" bessely="" functions="" (or="" matlab’s="" besselj="" and="" bessely)="" to="" de-="" termine="" numerical="" values="" for="" the="" smallest="" three="" eigenvalues="" for="" each="" problem,="" and="" plot="" the="" corresponding="" three="" eigenfunctions.="" 2="" 0321905652.pdf="" applied="" partial="" differential="" equations="" with="" fourier="" series="" and="" boundary="" value="" problems="" this="" page="" intentionally="" left="" blank="" applied="" partial="" differential="" equations="" with="" fourier="" series="" and="" boundary="" value="" problems="" fifth="" edition="" richard="" haberman="" southern="" methodist="" university="" boston="" columbus="" indianapolis="" new="" york="" san="" francisco="" upper="" saddle="" river="" amsterdam="" cape="" town="" dubai="" london="" madrid="" milan="" munich="" paris="" montreal="" toronto="" delhi="" mexico="" city="" sao="" paulo="" sydney="" hong="" kong="" seoul="" singapore="" taipei="" tokyo="" editor="" in="" chief:="" deirdre="" lynch="" cover="" designer:="" bruce="" kenselaar="" senior="" acquisitions="" editor:="" william="" hoffman="" manager,="" cover="" visual="" research="" and="" associate="" editor:="" brandon="" rawnsley="" permissions:="" jayne="" conte="" executive="" marketing="" manager:="" jeff="" weidenaar="" cover="" art:="" artida/fotolia="" marketing="" assistant:="" caitlin="" crain="" full-service="" project="" management:="" integra="" senior="" production="" project="" manager:="" beth="" houston="" composition:="" integra="" credits="" and="" acknowledgments="" borrowed="" from="" other="" sources="" and="" reproduced,="" with="" permission,="" in="" this="" textbook="" appear="" on="" the="" appropriate="" page="" within="" text.="" copyright="" c©="" 2013,="" 2004,="" 1998,="" 1987,="" 1983="" by="" pearson="" education,="" inc.="" all="" rights="" reserved.="" manufactured="" in="" the="" united="" states="" of="" america.="" this="" publication="" is="" protected="" by="" copyright,="" and="" permission="" should="" be="" obtained="" from="" the="" publisher="" prior="" to="" any="" prohibited="" reproduction,="" storage="" in="" a="" retrieval="" system,="" or="" transmission="" in="" any="" form="" or="" by="" any="" means,="" electronic,="" mechanical,="" photocopying,="" recording,="" or="" likewise.="" to="" obtain="" permission(s)="" to="" use="" material="" from="" this="" work,="" please="" submit="" a="" written="" request="" to="" pearson="" education,="" inc.,="" permissions="" department,="" one="" lake="" street,="" upper="" saddle="" river,="" new="" jersey="" 07458,="" or="" you="" may="" fax="" your="" request="" to="" 201-236-3290.="" many="" of="" the="" designations="" by="" manufacturers="" and="" sellers="" to="" distinguish="" their="" products="" are="" claimed="" as="" trademarks.="" where="" those="" designations="" appear="" in="" this="" book,="" and="" the="" publisher="" was="" aware="" of="" a="" trademark="" claim,="" the="" designations="" have="" been="" printed="" in="" initial="" caps="" or="" all="" caps.="" library="" of="" congress="" cataloging-in-publication="" data="" haberman,="" richard,="" applied="" partial="" differential="" equations="" :="" with="" fourier="" series="" and="" boundary="" value="" problems="" richard="" haberman.="" –="" 5th="" ed.="" p.="" cm.="" isbn-13:="" 978-0-321-79705-6="" (alk.="" paper)="" isbn-10:="" 0-321-79705-1="" (alk.="" paper)="" 1.="" differential="" equations,="" partial.="" 2.="" fourier="" series.="" 3.="" boundary="" value="" problems.="" i.="" title.="" qa377.h27="" 2013="" 515’.353–dc23="" 2012013033="" 10="" 9="" 8="" 7="" 6="" 5="" 4="" 3="" 2="" 1="" cw="" 16="" 15="" 14="" 13="" 12="" isbn-10:="" 0321797051="" isbn-13:="" 9780321797056="" to="" liz,="" ken,="" and="" vicki="" this="" page="" intentionally="" left="" blank="" contents="" preface="" xvii="" 1="" heat="" equation="" 1="" 1.1="" introduction="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 1="" 1.2="" derivation="" of="" the="" conduction="" of="" heat="" in="" a="" one-dimensional="" rod="" .="" .="" .="" .="" 2="" 1.3="" boundary="" conditions="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 11="" 1.4="" equilibrium="" temperature="" distribution="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 14="" 1.4.1="" prescribed="" temperature="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 14="" 1.4.2="" insulated="" boundaries="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 16="" 1.5="" derivation="" of="" the="" heat="" equation="" in="" two="" or="" three="" dimensions="" .="" .="" .="" .="" .="" .="" 19="" appendix="" to="" 1.5:="" review="" of="" gradient="" and="" a="" derivation="" of="" fourier’s="" law="" of="" heat="" conduction="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 30="" 2="" method="" of="" separation="" of="" variables="" 32="" 2.1="" introduction="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 32="" 2.2="" linearity="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 32="" 2.3="" heat="" equation="" with="" zero="" temperatures="" at="" finite="" ends="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 35="" 2.3.1="" introduction="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 35="" 2.3.2="" separation="" of="" variables="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 35="" 2.3.3="" time-dependent="" ordinary="" differential="" equation="" .="" .="" .="" .="" .="" .="" .="" .="" 37="" 2.3.4="" boundary="" value="" (eigenvalue)="" problem="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 38="" 2.3.5="" product="" solutions="" and="" the="" principle="" of="" superposition="" .="" .="" .="" .="" .="" 43="" 2.3.6="" orthogonality="" of="" sines="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 46="" 2.3.7="" formulation,="" solution,="" and="" interpretation="" of="" an="" example="" .="" .="" .="" 48="" 2.3.8="" summary="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 50="" appendix="" to="" 2.3:="" orthogonality="" of="" functions="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 54="" 2.4="" worked="" examples="" with="" the="" heat="" equation="" (other="" boundary="" value="" problems)="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 55="" 2.4.1="" heat="" conduction="" in="" a="" rod="" with="" insulated="" ends="" .="" .="" .="" .="" .="" .="" .="" .="" 55="" 2.4.2="" heat="" conduction="" in="" a="" thin="" insulated="" circular="" ring="" .="" .="" .="" .="" .="" .="" 59="" 2.4.3="" summary="" of="" boundary="" value="" problems="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 64="" 2.5="" laplace’s="" equation:="" solutions="" and="" qualitative="" properties="" .="" .="" .="" .="" .="" .="" .="" .="" 67="" 2.5.1="" laplace’s="" equation="" inside="" a="" rectangle="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 67="" 2.5.2="" laplace’s="" equation="" inside="" a="" circular="" disk="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 72="" 2.5.3="" fluid="" flow="" outside="" a="" circular="" cylinder="" (lift)="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 76="" 2.5.4="" qualitative="" properties="" of="" laplace’s="" equation="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 79="" vii="" viii="" contents="" 3="" fourier="" series="" 86="" 3.1="" introduction="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" 86="" 3.2="" statement="" of="" convergence="" theorem="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="" .="">